Comprehensive tests of the algorithm have been conducted demonstrating its accuracy and speed relative to Stokesian dynamics and spectral boundary integral computations. The speed of the algorithm is such that simulations of 4096 particles may be run on an IBM RS6000 workstation. Memory requirements are modest, and extension of the method to multiple-processors architectures is relatively straightforward. Results are discussed for both Monte Carlo and dynamic simulations for the sedimentation and rheology of concentrated hard-sphere suspensions.